A mathematical problem I can't seem to get Mathematica to auto solve.

[highdimensionman]

So to find log(n) in a given base  then n=b^log(n). so to find log n you find what b has to be to the power too in order for the sum to equal n.
Lets say for the function f1(n) that n=b+1/Powerroot(f1(n),b). So i'm trying to find the root dimension required for the sum to equal n quick.
The only reason I was studying this was because I wanted to see if there was any relationship with prime number distribution.

[polymath257]

So to find log(n) in a given base  then n=b^log(n). so to find log n you find what b has to be to the power too in order for the sum to equal n.

Your wording is confused: it would not be a sum, but a power.

In Mathemtica, the expression is Log[b,n].

Lets say for the function f1(n) that n=b+1/Powerroot(f1(n),b). So i'm trying to find the root dimension required for the sum to equal n quick.
The only reason I was studying this was because I wanted to see if there was any relationship with prime number distribution.

This seems confused.

In any case, it is known that the n^th prime, p_n is approximately n*Log[n], with the logarithm to the base e in the sense that the limit of the ration is 1 as n goes to infinity. This is related to the Prime Number Theorem (specifically, Rosser's theorem):

https://en.wikipedia.org/wiki/Prime_number_theorem

[Paleophyte]

Looks like it was a good night for falling down prime number theory rabbit holes. After wandering afoul of Euler and Reimann I bumped into number sieves and am now wondering how to calculate their efficiency. 2 takes care of 50% of all candidates, 3 eliminates 1 in 3, 5 does for 1 in 5... but what is the value of evaluating the remaining prime factors? Yes, I know you have to do it for clear and obvious reasons but I'm wondering how diminishing the returns are.

Put in what I hope are slightly less muddled terms: For any candidate k, where k is a finite but arbitrarily large positive integer, what is the probability (What is the convention for discussing primes and probabilities? Both seem to use p, which leads opens the doorway to some magnificent confusions.) that k is prime if k is not evenly divisible by the first n primes (2, 3, 5, 7, 11... pn) assuming that pn <<< sqrt(k).

I suspect that the answer to this is either laughably trivial or mind-breakingly difficult. There doesn't seem to be a lot of middle ground.

[HappySkeptic]

Looks like it was a good night for falling down prime number theory rabbit holes. After wandering afoul of Euler and Reimann I bumped into number sieves and am now wondering how to calculate their efficiency. 2 takes care of 50% of all candidates, 3 eliminates 1 in 3, 5 does for 1 in 5... but what is the value of evaluating the remaining prime factors? Yes, I know you have to do it for clear and obvious reasons but I'm wondering how diminishing the returns are.

Put in what I hope are slightly less muddled terms: For any candidate k, where k is a finite but arbitrarily large positive integer, what is the probability (What is the convention for discussing primes and probabilities? Both seem to use p, which leads opens the doorway to some magnificent confusions.) that k is prime if k is not evenly divisible by the first n primes (2, 3, 5, 7, 11... pn) assuming that pn <<< sqrt(k).

I suspect that the answer to this is either laughably trivial or mind-breakingly difficult. There doesn't seem to be a lot of middle ground.

I programmed Sundarum's Sieve when in grade 11.  It worked pretty well, but I'm not sure what the best ones are.

[polymath257]

Looks like it was a good night for falling down prime number theory rabbit holes. After wandering afoul of Euler and Reimann I bumped into number sieves and am now wondering how to calculate their efficiency. 2 takes care of 50% of all candidates, 3 eliminates 1 in 3, 5 does for 1 in 5... but what is the value of evaluating the remaining prime factors? Yes, I know you have to do it for clear and obvious reasons but I'm wondering how diminishing the returns are.

Put in what I hope are slightly less muddled terms: For any candidate k, where k is a finite but arbitrarily large positive integer, what is the probability (What is the convention for discussing primes and probabilities? Both seem to use p, which leads opens the doorway to some magnificent confusions.) that k is prime if k is not evenly divisible by the first n primes (2, 3, 5, 7, 11... pn) assuming that pn <<< sqrt(k).

I suspect that the answer to this is either laughably trivial or mind-breakingly difficult. There doesn't seem to be a lot of middle ground.

Use capital P for probability and lower case p for primes.

There is an extension of the prime number theorem for primes appearing in an arithmetic progression that might be useful for the probability you are wanting.

[Jehanne]

How to map a uniform probability space on an infinite set?

[polymath257]

How to map a uniform probability space on an infinite set?

The problem is that the set of natural numbers is countable and so does not have a uniformly distributed measure on it.

That said, the density of a set is often a good alternative. If A is a set, consider the ratio #{x\in A: x<=n} / n as n-->infinity.

While the limit doesn't always exist, this has some of the properties we expect from probabilities.